The formulation is applied to two cases. The first considers an underexpanded jet at low supersonic Mach number, modelled by a canonical slowly-diverging flow. The solution procedure is shown in detail for two frequencies, including the definition of the initial solution to be marched downstream. The modulation characteristics of the resulting axisymmetric wavepacket are explored, and the initial results suggest that the modulation can be very sensitive to the different modes supported by the flow at a given frequency. Overall, some characteristics of the flow structures obtained using PFE are similar to results in the literature, which provided a first validation of the method. After that, the same methodology was applied to predict the most energetic (non-screeching) helical structures from an LES of an overexpanded jet. For that, a slightly more complex mean flow model was chosen to provide a better comparison with the LES data. Using such a model, PSE was able to predict the overall behaviour of the shock-cell structure, especially its leading wavenumber and the shock spacing variation further downstream. Using this solution, PFE modes were computed and compared to pressure SPOD modes for azimuthal wavenumber \(m=1\) . An overall good agreement was obtained in the predictions, especially closer to the centreline of the jet, suggesting that the method is able to capture the underlying physical mechanisms for the generation of these structures: the extraction of energy from the mean flow by the Kelvin-Helmholtz mode, and a redistribution of energy to modulation wavenumbers due to the interaction with the shock-cell structure, as predicted by [1]}. The model outperforms the classic PSE method, providing a good first approximation of the wavepackets supported by the flow, despite its several limitations. No experimental input is needed for the predictions, but a representative ideally expanded mean flow must be provided. While zero-frequency PSE is used to predict the overall behaviour of the shock-cell structure, a decomposition of a experimentally obtained shock-containing mean flow following the same structure as in the present case could also be used as input in the analysis.
